34 III METRIC SPACES

(3.5.1) Any open ball is an open set.

For if Jt£B(a;r), then d(a, x) < r by definition; hence the relation
d(x, y)<r - d(a, x) implies d(a, y) < d(a, x) + d(x, y) < r, which proves the
inclusion B(x; r - d(a, x)) c E(a; r).

(3.5.2) The union of any family (AA)AeL of open sets is open.
For if x £ A for some fi e L, then there is r > 0 such that

AeL

For instance, in the real line R, any interval ]a, + oo[ is open, being the
union of the open sets ]a, x[ for all x > a. Similarly, ] -~ oo, a[ is open.

(3.5.3) The intersection of a finite number of open sets is open.

It is enough to prove that the intersection of two open sets Al5 A2 is
open, and then to argue by induction. If x e At n A2 , there are ^ > 0,
r2 > 0 such that B(JC; r^) e A1? E(x; r2) c A2; clearly if r = min^, r2),
B(x; r) c: A! n A2.

In general, an infinite intersection of open sets is no longer open; for
instance the intersection of the intervals ] — 1/n, l/n[ in R is the one point
set {0}, which is not open by (2.2.16). However:

(3.5.4) In a discrete space any set is open.

Due to (3.5.2), it is enough to prove that a one point set {a} is open.
But by definition, {a} = R(a; 1/2), and the result follows from (3.5.1).

6. NEIGHBORHOODS

If A is a nonempty subset of E, an open neighborhood of A is an open
set containing A ; a neighborhood of A is any set containing an open neigh-
borhood of A. When A - {x}, we speak of neighborhoods of the point x
(instead of the set {x}). i/, 4ii ,ii*l i% a A of !*